FULLY CONVEX NORMED LINEAR SPACES.

Foi any integer k > 2 and for i = 1, 2, 3, 4, let us consider the following conditions concerning X: (F.k.C.i) Every (k.i)-sequence {xnj in X is a Cauchy sequence. (W.F.k.C.i) Every (k.i)-sequence {xnj in X is weakly convergent.' Condition (F.2.C.2) has been previously considered by V. Smulian.2 THEOREM 1. For any fixed integer k > 2 and for any normed linear space X, the four conditions (F.k.C.i), 1 < i < 4, are mutually equivalent. Also, the four conditions (W.F.k.C.i), 1 < i < 4, are mutually equivalent. Proof: To see that (F.k.C.1) =+ (F.k.C.2) and (W.F.k.C.1) =+(W.F.k.C.2), it suffices to show that for any (k.2)-sequence {[xn in a normed linear space, we can find a sequence { an) of real numbers with lim an = 1 such that { a1xn} is a -0 CO